Introduction to Density Functional Theory · symmetry, the jellium model has a constant electron density. Hence, with the correct jellium , we could calculate the energy of any material

Introduction to DFTPHYS824: Nanophysics and Nanotechnology

Introduction to Density Functional Theory

Branislav K. NikolićDepartment of Physics and Astronomy, University of Delaware,

Newark, DE 19716, U.S.A.http://wiki.physics.udel.edu/phys824


Why Study DFT: Shows the Limitations of Simple Models of Solids

3t− 2t− 1t− 0 2t1t 3tEnergy

Den

sity

of S

tate

s

holes electrons0

1


Why Study DFT: Makes Possible Search for New Materials In Silico


Why Study DFT: When Combined with Machine Learning It Can Predict New Materials

Example: PRL 108, 068701 (2012) identifies via high-throughput DFT+GW calculations of ∼260 generalized IpIIIqVIr chalcopyrite materials, out of which 20 are highly efficient for photovoltaic applications (including best known as well as previously unrecognized ones).


Why Study DFT: Shows the Limitations of Simple Models of Magnetic Materials


Why Study DFT: When Combined with Quantum Transport It Can Predict Realistic New Devices

PRB 85, 184426 (2012)

NiNi Grn

PRL 99, 176602 (2007)


Quantum-Mechanical Many-Body Problem for Electrons in Solids

To calculate material properties, one has to take into account three terms: The kinetic energy favoring the electrons to move through the crystal (blue), the lattice potential of the ions (green), and the Coulomb interaction between the electrons (red). Due to the latter, the first electron is repelled by the second, and it is energetically favorable to hop somewhere else, as depicted. Hence, the movement of every electron is correlated with that of every other, which prevents even a numerical solution.

Hamiltonian of electrons in solid after Born-Oppenheimer approximation:

Many traditional approaches to solving difficult many-body problem begin with the Hartree–Fock approximation, in which many-particle wavefunction is approximated by a single Slater determinant of orbitals (single-particle wavefunctions) and the energy is minimized. These include configuration interaction, coupled cluster, and Møller-Plesset perturbation theory, and are mostly used for finite systems, such as molecules in the gas phase. Other approaches use reduced descriptions, such as the density matrix or Green function, but leading to an infinite set of coupled equations that must somehow be truncated.


Density Functional Theory Approach to Quantum Many-Body Problem

Local Density Approximation (LDA) is an approximation which allows to calculate material properties but which dramatically simplifies the electronic correlations: Every electron moves independently, i.e., uncorrelated, within a time-averaged local density of the other electrons, as described by a set of single-particle Kohn-Sham equations whose solutions (“orbitals”) are used the built the density.

DFT conceptual structure:

DFT in computational practice:

Conventional quantum-mechanical approach to many-body systems:

Knowledge of electron density implies knowledge of the wave function and the potential, and hence of all other observables. Although this sequence describes the conceptual structure of DFT, it does not really represent what is done in actual applications of it, which typically proceed along rather different lines, and do not make explicit use of many-body wave functions


Computational Complexity: Many-Body Wave Function vs. Electron Density vs. KS Orbitals


DFT is Based on the Exact Hohenberg-Kohn Theorem

1. The nondegenerate ground-state (GS) wave function is a unique functional of the GS density:

2. The GS energy, as the most important observable, has variational property:

3. Kinetic energy and electron-electron interaction energy are universal (system independent) functionals of electron density:

while non-universal potential energy in external field is obtained from:

4. If external potential is not held fixed, the functional becomes universal: the GS density determines not only the GS wave function , but, up to an additive constant, also the potential . CONSEQUENCE:


Two Step Formal Construction of the Energy Density Functional

While this construction proves the Hohenberg–Kohn theorem, we did not actuallygain anything: to obtain the functional we have to calculate the expectation value for complicated many-body wave functions .


Functionals and Functional Derivatives Function vs. Functional:

Standard Derivative vs. Functional Derivative:

Useful formula:

function maps number to a number

functional maps whole function to a number; example:

is the same as

the integral arises because the variation in

the functional F is determined by variations

in the function at all points in space

Example from classical mechanics:


Practical DFT: Kohn-Sham Equations Only the ionic (external for electrons) and the Hartree (i.e., classical Coulomb) potential energy can be expressed easily through the electron density:

Using the kinetic energy functional, all of the difficulty of electron-electron interactions is absorbed into the exchange-correlation term:

Although the exact form of is unknown, an important aspect of DFT is that the functional does not depend on the material investigated, so that if we knew the DFT functional for one material, we could calculate all materials by simply adding .

To calculate the ground-state energy and density, we have to minimize:

Kohn-Sham: Electron density expressed in terms of auxiliary one-particle wave function

Kohn-Sham: The final result is a set of one-particle Schrödinger equations

describing single electrons moving in time-averaged potential of all electrons


Kohn-Sham Equations in the Local Density Approximation (LDA)

The one-particle Kohn–Sham equations, in principle, only serve the purpose of minimizing the DFT energy, and have no physical meaning.

If we knew the exact , which is non-local in , we would obtain the exact ground state energy and density. In practice, one has to make approximations to such as the LDA:

NOTE: The kinetic energy in KS equations is that of independent (uncorrelated) electrons. The true kinetic

energy functional for the many-body problem is different. We hence have to add the difference between the true

kinetic energy functional for the many-body problem and the above uncorrelated kinetic energy to , so that all

many-body difficulties are buried in .

is typically calculated from the perturbative solution or the numerical simulation of the jellium model which is defined by . Owing to translational symmetry, the jellium model has a constant electron density

. Hence, with the correct jellium , we could calculate the energy of any material with a constant electron density exactly. However, for real materials is varying, less so for s and p valence electrons but strongly for d and f electrons.


Kohn-Sham “Quasiparticles” Dragging Exchange-Correlation Hole

Wave function of two non-interacting electrons

Probability to find electron i in some small volume dri while electron j is in some small volume drj

Effective charge density seen by an electron of a given spin orientation

in a non-interacting electron gas


Practical DFT: PseudopotentialsIf we knew the DFT functional for one material, we could treat all materials by simply adding ionic potential which is typically approximated by pseudopotentials.

E. K

axir

as, A

tom

ic a

nd e

lect

roni

c st

ruct

ure

of s

olid

s(C

UP,

200

3)


Solving Kohn-Sham Equations via Self-Consistent Loop: Computational Codes


Examples: DFT Applied to 0D Systems of Quantum Chemistry

Two non-interacting fake electrons (doubly occupying 1s orbital) sitting in this potential have precisely the

same density as the interacting electrons in He atom. If we can

figure out some way to approximate this potential accurately, we have a

much less demanding set of equations to solve than those of the

true system.

http://dft.uci.edu/dftbook.html

LDA greatly improves on Hartree-Fock, but it typically overbinds by about 1 eV, which is too

inaccurate for quantum chemistry, but sufficient for many solid-state calculations


Beyond DFT: GW and DFT+GW for Weakly-Correlated Systems

H0 is the (Hermitian) Hamiltonian of a single particle in external potential of ions and classical Hartree potential

In many cases there is an almost complete overlap between the QP and the KS wavefunctions, and the full resolution of the QP equation may be

circumvented by computing the quasiparticle energy using a first-order perturbation of the KS energy.

The total quasiparticle Hamiltonian is non-Hermitian!

Green function screened Coulomb interaction


DFT vs. GW vs. Experiment for Bi2Te3

Using moderate energy, ARPES can observe electronic states that are buried deep below the surface if only

an evanescent tail of the wave function is present within the mean free path of the photoelectrons


DFT+GW Fitted with Slater-Koster TBH Using pz,dyz,dzx Orbitals per C Atom

Slater-Koster hopping parameters (in eV)


Alternative Ab Initio TBH for Graphene with Single pz Orbital per C Atom and 3NN Hoppings


DFT fitted with Slater-Koster TBH for Monolayer MoS2


Trouble with Simplistic Hamiltonians for Van der Waals Heterostructures

Scie

nce 34

6, 4

48 (2

014)

CONTROVERSYNat. Commun. 8, 14552 (2017) J. Phys. Mater. 1, 0150061 (2018)

PRB 79, 035304 (2009)

Sci. Adv. 4, eaaq0194 (2018)


Resolution of the Trouble with Ab Initio TBHs

Edge

cur

rets

for

TBH

J. Phys. Mater. 1, 0150061 (2018)


Construction of Ab Initio TBH in the basis of Maximally Localized Wannier Functions (MLWF)

MLWF: Faithfully retain the overlap matrix elements and their phases, the orbital

character of the bands and the accuracy of the original DFT calculations, but at the

computational cost of TBH

WF Bloch state

multiband generalized WF (localization in R gives Bloch smoothness in k)

chosen to minimize quadratic spread

G/hBN

arXiv:1508.07735

PRB

92, 2

0510

8 (2

015)

Amplitude isosurface contours of MLWF for graphene is split into the dominant pz

component with m = 0 angular momentum, plus two additional components with m = 3

and m = 6 angular momentum (which is defined up to module 3 due to restoration of threefold rotation symmetry at the position of C atom)

whose radius increases with m.

Wannier TBH vs. fitted Slater-Koster TBH

Examples: Graphene and Graphene/hBN


Beyond DFT: DMFT and DFT+DMFT for Strongly Correlated Materials

The success of LDA shows that this treatment is actually sufficient for many materials, both for

calculating ground state energies and band structures, implying that electronic correlations are

rather weak in these materials. But, there are important classes of materials where LDA fails, such as transition metal oxides or heavy fermion systems. In these materials the valence orbitals are the 3d and 4f orbitals. For two electrons in these orbitals the distance is particularly short, and electronic

correlations particularly strong.

Many such transition metal oxides are Mott insulators, where the on-(lattice-)site Coulomb

repulsion U splits the LDA bands into two sets of Hubbard bands. One can envisage the lower Hubbard band as consisting of all states with one electron on every lattice site and the upper Hubbard band as those states where two electrons are on the same lattice site. Since it costs an energy U to have two

electrons on the same lattice sites, the latter states are completely empty and the former completely

filled with a gap of size U in-between. Other transition metal oxides and heavy fermion systems

are strongly correlated metals, with heavy quasiparticles at the Fermi energy, described by an

effective mass or inverse weight m/m0=1/Z»1.


Beyond DFT: Coulomb Blockade in Nanostructures

H. v

an H

oute

n, C

. W. J

. Bee

nakk

er, a

nd A

. A. M

. Sta

ring

,co

nd-m

at/0

5084

54Ensslin Lab, A

PL 92, 012102 (2008)


DFT References for Beginners

Documents

Introduction to Density Functional Theory · symmetry, the jellium model has a constant electron density. Hence, with the correct jellium , we could calculate the energy of any material